Introduction to the Mathematics of Infectious Diseases Vrushali A Bokil Department of Mathematics Oregon State University, Corvallis, Oregon 2008 REU Program in Mathematics Oregon State University, August 1, 2008 Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 1 / 36
Outline The Beginnings of Mathematical Epidemiology 1 Importance of Mathematical Modeling 2 3 Basic Compartmental Deterministic Models Basic Ideas and Assumptions The SIS, SIR, SEIR Models The SIR Epidemic Model 4 Numerical Simulations Conditions for an Epidemic The Basic Reproduction Number 5 The SIR Endemic Model Computing Endemic Equilibrium Numerical Example Limitations 6 Conclusions 7 Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 2 / 36
The Beginnings of Mathematical Epidemiology Bernoulli: 1760 1 Daniel Bernoulli formulated and solved a model for smallpox in 1760 Using his model, he evaluated the effectiveness of inoculating of healthy people against the smallpox virus. Hamer: 1906 2 Hamer formulated and analyzed a discrete time model in 1906 to understand the recurrence of measles epidemics. Ross: 1911 3 Ross developed differential equation models for malaria as a host-vector disease in 1911. He won the second nobel prize in medicine Kermack and McKendrick: 1926 4 Extended Ross’s models. Obtained the epidemic threshold results. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 3 / 36
Main Scope of Mathematical Modeling in Epidemiology Bailey We need to develop models that will assist the decision making process by helping to evaluate the consequences of choosing one of the alternative strategies available. Thus, mathematical models of the dynamics of a communicable disease can have a direct bearing on the choice of an immunization program, the optimal allocation of scarce resources, or the best combination of control or eradication techniques. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 4 / 36
Main Scope of Mathematical Modeling in Epidemiology Okubo A mathematical treatment is indispensable if the dynamics of ecosystems are to be analyzed and predicted quantitatively. The method is essentially the same as that used in such fields as classical and quantum mechanics, molecular biology and biophysics... One must not be enamoured of mathematical models; there is no mystique associated with them... physics and mathematics must be considered as tools rather than sources of knowledge, tools that are effective, but nonetheless dangerous if misused. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 5 / 36
Why do Mathematical Modeling? Clarification The model formulation process clarifies assumptions, variables, and parameters Models provide conceptual results such as thresholds: basic reproduction numbers, contact numbers, and replacment numbers. Thresholds are critical values for quantities such as population size or vector density that must be exceeded in order for an epidemic to occur. Simulated Experimentation Mathematical models and computer simulations are useful experimental tools for building and testing theories, assesing quantitative conjectures, answering specific questions, determining sensitivities to changes in parameter values and estimating key parameters from data Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 6 / 36
Why do Mathematical Modeling? (cont) Complex Transmission Interactions Transmission Interactions in a Population are very complex: It is difficult to comprehend the large scale dynamics of disease spread. Understanding these interaction characteristics can lead to better approaches to decreasing the transmission of diseases. Mathematical models are used in comparing, planning implementing, evaluating, and optimizing various detection, prevention, therapy and control programs. Realistic Experimenting is Impossible Experiments with Infectious disease spread in human populations are often impossible, Unethical or expensive. Data is sometimes available after the fact from naturally occurring epidemics and is incomplete due to under reporting. Epidemiological modeling can contribute to the design and analysis of episdemiological surveys, suggest crucial data that should be collected, identify trends, make general forecasts, and estimate the uncertainty in forecasts. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 7 / 36
Basic Ideas and Assumptions Deterministic Models For a given model structure, chosen (fixed) parameter values, and particular initial conditions, the model produces the same output each time it is simulated. This is one way of simulating infectious disease dynamics Reasonable if the sizes of the populations are large. Compartmental Models Populations under study are divided into compartments. Assumptions are made about the nature and time rate of transfer from one compartment to another. Rates of transfer between compartments are expressed mathematically as derivatives with respect to time of the sizes of the compartments. Models are systems of differential equations Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 8 / 36
Basic Ideas and Assumptions (cont) Crucial Idea!!! The derivative function represents the rate of change of that function Assumptions The community size is constant over the duration of the epidemic and is a large number, call it N . The infection is transmitted primarily by person-to person contacts (e.g., measles) Individuals are homogeneous and mix uniformly. Transmission rates, removal rates are constant. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 9 / 36
Basic Compartmental Deterministic Models SIS, SIR, SEIR SIS Model S I SIR Model S I R SEIR Model S E I R Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 10 / 36
Basic Compartmental Deterministic Models SIS, SIR, SEIR SIS Model S I SIR Model S I R SEIR Model S E I R The choice of which compartments to include in a model depends on the characteristics of the particular disease being modeled and the purpose of the model. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 10 / 36
The SIR Epidemic Model Additional Assumptions Ignore demography, i.e., births and deaths The SIR Epidemic Model Transmission Recovery S I R Compartments Susceptibles ( S ): Individuals susceptible to the disease Infectious ( I ): Infected Individuals able to transmit the parasite to others Recovered ( R ): Individuals that have recovered, are immune or have died from the disease and do not contribute to the transmission of the disease S = S ( t ) , I = I ( t ) , R = R ( t ) and N = S ( t ) + I ( t ) + R ( t ) Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 11 / 36
The Basic SIR Epidemic Model: Contact Rates The SIR Epidemic Model β α S I R Let s = S / N , i = I / N and r = R / N . Let β = Average number of adequate contacts (i.e., contacts sufficient for transmission) of a person per unit time. β I N = β i = Average number of contacts with infectives per unit time of one susceptible. � β I � S = β Nis = Number of new cases per unit time due to the N S = Ns susceptibles. ( Horizontal Incidence ) Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 12 / 36
Assumptions of the SIR Epidemic Model Consider the “cohort” of members who were all infected at one time and let u ( q ) denote the number of these who are still infective q units after having been infected. If a fraction α of these leave the infective class in unit time, then u ′ = − α u whose solution is u ( q ) = u ( 0 ) e − α q The fraction of infectives remaining infective q time units after having become infective is e − α q The length of the infective period is distributed exponentially with mean � ∞ e − α q dq = 1 /α 0 Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 13 / 36
The Basic SIR Epidemic Model: Waiting Times The SIR Epidemic Model β α S I R The deterministic SIR epidemic model for this process is dS dt = − β I S N dI dt = β I S N − α I dR dt = α I The parameters of the model are β = the transmission rate (effective contact rate) α = the recovery or removal rate Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 14 / 36
The Basic SIR Deterministic Epidemic Model The SIR Epidemic Model β α S I R Let s = S / N , i = I / N and r = R / N . Dividing the equations for S , I and R by N we get the deterministic SIR epidemic model for this process in the form ds dt = − β si di dt = β si − α i dr dt = α i Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 15 / 36
Assumptions of the SIR Epidemic Model The SIR Epidemic Model β α S I R An average infective makes contact sufficient to transmit infection with β others in unit time. A fraction α of infectives leave the infective class in unit time. There is no entry or departure from the population except possibly through death from the disease. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 16 / 36
The Basic SIR Deterministic Epidemic Model: A Numerical Simulation Example 1 Initial values are: i ( 0 ) = 0 . 001, s ( 0 ) = 0 . 999 , r ( t ) = 0, Parameter values are: β = 0 . 3, α = 0 . 1. SIR Model Infectives in SIR Model 1 0.3 Fraction of Infectives 0.8 s, i, and r 0.2 s 0.6 i 0.4 r 0.1 0.2 0 0 0 50 100 0 50 100 Time t (Days) Time t (Days) Model predicts that there is an epidemic. Bokil (OSU) Mathematical Epidemiology REU-Math, 2007 17 / 36
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